CS 473: COMPILER DESIGN
1 LOOPS AND DOMINATORS
2 Loops in Control-flow Graphs
• Most programs spend most of their time in loops, so if we want to optimize, loops are a good place to look
• Like other optimizations, loop optimizations are best applied to a control flow graph IR – Other opts may create opportunities for loop opts and vice versa, so it makes sense to alternate between them
• Loops may be hard to recognize at the CFG IR level – Many kinds of loops: while, do/while, for, loops with break/continue, goto… all of which turn into some combination of comparisons, labels, jumps, and body code
• Problem: How do we identify loops in the control flow graph?
3 Definition of a Loop
• A loop is a set of nodes in the control flow graph such that: • There is a single distinguished entry point called the header
loop • Every node is reachable nodes from the header & the header is reachable from every node header – A loop is a strongly connected component exit node
• No edges enter the loop except to the header
• Nodes with outgoing edges are called loop exit nodes 4 Nested Loops
• Control-flow graphs may contain many loops • Loops may contain other loops:
Loop Nesting Tree:
5 Loop Analysis • Goal: Identify the loops and nesting structure of the CFG
• Loop analysis is based on the idea of dominators: node A dominates node B if the only way to reach B from the start node is through node A – The header of a loop dominates all the nodes in the loop
• An edge in the graph Back Edge is a back edge if the target node dominates the source node
• Every loop contains at least one back edge
6 Dominator Dataflow Analysis
• We can define Dom[n] as a forward dataflow analysis. • Using our usual framework:
• “A node B is dominated by another node A if A dominates all of the predecessors of B.”
– in[n] := ∩n’∈pred[n]out[n’] • “Every node dominates itself.” – out[n] := in[n] ∪ {n}
• We start with every node in all of the sets, and in each iteration remove those that don’t dominate all of the node’s predecessors • At the end, out[n] will be the set of nodes that dominate n
7 8 Dominator Trees
• Result: for each node, a set of nodes that dominate it • Each node has one immediate dominator, the closest node that dominates it • We can draw a tree where each node’s parent is its immediate dominator
CFG: Dominator Tree:
4 4
5 6 5 6
8 8
9 Dominator Trees
• Result: for each node, a set of nodes that dominate it • Each node has one immediate dominator, the closest node that dominates it
CFG: Dominator Tree:
1 1
2 2
3 4 3 4
5 6 5 6
7 8 7 8
9 0 9 0 10 Completing Loop Analysis
• Dominator analysis identifies back edges: edges n → h where h dominates n • Each back edge has a natural loop: h – h is the header
– All nodes reachable from h that also reach n n without going through h are in the loop
1
2
3 4
5 6
7 8
9 0 11 Completing Loop Analysis
• Dominator analysis identifies back edges: edges n → h where h dominates n • Each back edge has a natural loop: h – h is the header
– All nodes reachable from h that also reach n n without going through h are in the loop
• It’s convenient for each loop to have a unique header, so if two loops share the same header, h we merge them n m
• If all the nodes in one loop are also in another loop, the first loop is nested inside the second
12 Example Natural Loops
1 Loop Nesting Tree:
2
3 4
5 6
7 8
9 0
Natural Loops
13 14 Using Loop Information
• Loop nesting depth plays an important role in optimization heuristics – Deeply nested loops pay off the most for optimization
• Need to know loop headers / back edges for doing: – loop invariant code motion (remove code from loop if it’s the same every time) – loop unrolling (execute n iterations of the loop at once) – and lots more!
• Dominance information also plays a role in converting to Static Single Assignment form
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